which has not been considered in CoMP transmission in prior works [71].
2.3
Optimal solutions via the BnC method
2.3.1
The continuous relaxation of the big-M formulation
The formulated JNOB problem (2.12) and other general MISOCPs, can be solved using the convex continuous relaxation based BnC method [67–69, 81, 82]. The continuous relaxation of a MISOCP is the SOCP obtained by relaxing all the integer constraints. The convex continuous relaxation of the formulated JNOB problem in (2.12) can be expressed as the fol- lowing SOCP, which is referred to as the big-M continuous relaxation (BMC) in the sequel:
Φ(BMC) , min {wk,l,ak,l,bl} f {ak,l}, {bl}, {wk,l} (2.13a) s.t. (2.9a): Im{hHkwk} = 0, ∀k ∈ K (2.9b): hHkW, σk 2 ≤ γkRe{hHk wk}, ∀k ∈ K (2.12b): v u u t K X k=1 kwk,lk22 ≤ bl q Pl(MAX),∀l ∈ L (2.12c): kwk,lk2 ≤ ak,l q Pl(MAX),∀k ∈ K, ∀l ∈ L (2.12d):ak,l ≤ bl,∀k ∈ K, ∀l ∈ L (2.12e): L X l=1 ak,l ≥ 1, ∀k ∈ K 0≤ ak,l ≤ 1, 0 ≤ bl≤ 1, ∀k ∈ K, ∀l ∈ L (2.13b) where the variables{ak,l, bl,∀k ∈ K, ∀l ∈ L}, originally constrained to take integer values in (2.12f), are relaxed to continues variables in the closed interval[0, 1] in (2.13b).
We assume that the point characterized by the parameter tuplew(BMC)k,l , a(BMC)k,l , b(BMC)l , ∀k ∈ K, ∀l ∈ L is an optimal (not necessarily unique) solution of the SOCP in (2.13). Since the objective function in (2.13a) is minimized, we can easily prove by contradicting
argument the following properties: L X l=1 b(BMC)l ≥ 1 (2.14) K X k=1 a(BMC)k,l ≥ b (BMC) l , ∀l ∈ L. (2.15)
Assume that the pointw(BMI)k,l , a(BMI)k,l , b(BMI)l ,∀k ∈ K, ∀l ∈ L is an optimal (unneces- sarily unique) solution of the JNOB problem in (2.12). We show next that the optimal ob- jective value of the continuous relaxation in (2.13) is strictly smaller than that of the JNOB problem (2.12) for practical systems with CoMP transmission. Towards this end, we first present the necessary conditions for which the JNOB problem (2.12) and the associated con- tinuous relaxation (2.13) achieve the same optimal objective value, as summarized in the following theorem.
Theorem 2.1 (Necessary Conditions). If the JNOB problem in (2.12) and the associated
continuous relaxation in (2.13) achieve the same optimal objective value, i.e., if Φ(BMI) = Φ(BMC), the following conditions must hold:
K X j=1 a(BMI)j,l = K X j=1
a(BMI)j,m = 1, if a(BMI)k,l = a(BMI)k,m = 1,
for somek∈ K, l 6= m, ∀l, m ∈ L. (2.16)
That is if thelth BS cooperates with the mth BS to serve the kth MS, then the lth and the mth
BSsexclusively serve thekth MS in the case that Φ(BMI) = Φ(BMC). Proof 2.1. Please refer to Appendix A.1 for the proof.
We know from Theorem 2.1 that the special case of Φ(BMI) = Φ(BMC) may happen if each of the cooperating BSs (i.e., the BSs that jointly serve MSs in CoMP transmission) serves exclusively a single MS. However, in practical cellular networks employing CoMP transmission, the necessary conditions in (2.16) generally do not hold, since cooperating BSs usually serve multiple MSs to suppress ICI and to improve spectral efficiency. As a result, the following corollary represents a direct application of Theorem 2.1.
Corollary 2.1. In cellular networks with multiple MSs served jointly by cooperating BSs in
2.3. Optimal solutions via the BnC method 21
smaller than that of the JNOB problem (2.12), i.e.,
Φ(BMC)< Φ(BMI). (2.17)
We further observe that we can setak,l = 1 and bl = 1,∀k ∈ K, ∀l ∈ L, for testing the feasibility of the JNOB problem (2.12). If the JNOB problem (2.12) is feasible, then a fully connected network is a feasible network topology. This suggests that if the SOCP in (2.13) is feasible, e.g., with a feasible solution given by the parameter tuplew(FES)k,l , a(FES)k,l , b(FES)l , ∀k ∈ K, ∀l ∈ L , then the point w(FES)k,l , ak,l = 1, bl = 1,∀k ∈ K, ∀l ∈ L
is a feasible solution of the JNOB problem (2.12). As a result, the JNOB problem (2.12) is feasible if and only if the associated continuous relaxation in (2.13) is feasible.
2.3.2
Overview of the BnC method and the solver CPLEX
Thanks to the vast advancement of parallel computing, the convex continuous relaxation based BnC method [67–69, 81, 82] is widely adopted for solving MISOCPs and is imple- mented in the commercial solvers, e.g., in IBM ILOG CPLEX [81]. We present here a brief overview of the continuous relaxation based BnC method, based on the JNOB problem in (2.12) and the associated continuous relaxation in (2.13).
The BnC method is a combination of the branch-and-bound (BnB) procedure and the cutting plane (CP) algorithm [67–69, 81, 82]. As in the BnB procedure, a binary search tree that consists of nodes is constructed in the BnC algorithm, as shown in Fig. 2.2. Each node on the search tree represents the continuous relaxation, which is a SOCP as that of the SOCP in (2.13), of a subproblem resulted from fixing one or more binary integer variables in the original MISOCP (2.12) [67–69, 81, 82]. The BnC search tree is initialized with one node, e.g., the root node that represents the continuous relaxation in (2.13) of the JNOB problem (2.12), as illustrated in Fig. 2.2. If the solution of the SOCP represented by a node is not integer-feasible, the BnC procedure chooses one relaxed binary variable that is not integer-valued in the solution to perform a branching step. As a result, parting from the current node, two subproblems are created by fixing the chosen variable to be one and zero, respectively, which are represented by two descendant nodes of the current node (cf. Fig. 2.2). This branching process is carried out recursively at each node on the BnC search tree. Considering a minimization problem such as the JNOB problem in (2.12), a node and its descendants (i.e., the subtree rooted at that node) can be removed from the BnC search tree if one of the following pruning conditions is satisfied [67–69, 81, 82]:
(C2) The solution of the continuous relaxation at the node is integer-feasible (deleting the node and recording the integer-feasible solution).
(C3) The optimal objective value of the continuous relaxation at the node is larger than that of the incumbent solution (deleting the node and its descendants). The incumbent solution is the best-known integer-feasible solution, i.e., the one with the smallest objective value among the recorded integer-feasible solutions.
The pruning conditions (C1) – (C3) are also displayed in Fig. 2.2.
,
[0,1],
[0,1],
,
k l la
∈
b
∈
∀ ∈k
K
∀ ∈l
L
1,11
a
=
2,11
a
=
1,10
a
=
2,10
a
=
2,11
a
=
a
2,1=0
3,11
a
=
a
3,1=0
a
3,1=1
a
3,1=0
a
3,1=1
a
3,1=0
1,21
a
=
a
1,2=0
Figure 2.2: Illustration of the BnC solution process and the pruning conditions.
Further, we know from the pruning conditions (C1) – (C3) that the size of the search tree and the computational complexity of the BnC algorithm depend critically on the formulation of the MISOCP, as well as the tightness of the continuous relaxation of the sub-MISOCP at each node [67–69, 81, 82]. Throughout this thesis, the tightness of a continuous relaxation refers to the absolute gap between the optimal objective value of a MISOCP and that of the associated continuous relaxation. For instance, the term Φ(BMI) − Φ(BMC) represents the tightness of the continuous relaxation in (2.13). In this sense, a smaller gap of Φ(BMI) − Φ(BMC) corresponds to a tighter continuous relaxation in (2.13).
The solution of the continuous relaxation at a node provides a local lower bound (LLB) on the optimal objective value of the corresponding sub-MI-SOCP at that node and its de- scendants. The LLBs are important for pruning nodes and reducing the size of the search tree according to the pruning condition (C3). The minimum among the LLBs of the nodes repre- sents a global lower bound (GLB) of the optimal objective value of the JNOB problem (2.12). The GLB is important for computing optimality certificates (see Section 2.5.1) [67–69, 81, 82]. In the BnC procedure, the GLB on the optimal objective value of the original MIS- OCP (2.12) is successively improved due to the branching operations on some of the relaxed binary variables. Hence, the optimality certificate is eventually obtained as the branching
2.3. Optimal solutions via the BnC method 23
process continues if the runtime allows. The standard BnC method is commonly imple- mented with parallel processing threads as in, e.g., the commercial MIP solver IBM ILOG CPLEX [81].
During the tree-searching process of the BnC algorithm, cuts may be generated at each node. Cuts are linear (and/or convex) constraints added to a MISOCP to reduce the size of the feasible set of the associated continuous relaxations [67–69,81,82]. That is, cuts are con- straints that are redundant (i.e., not affecting the feasible set) for the original MISOCPs, but they reduce the size of the feasible sets of the associated continuous relaxations, as illustrated in Fig. 2.3. For instance, the following constraints, i.e., the constraints in Eq. (2.12e):
(2.12e): L X
l=1
ak,l ≥ 1, ∀k ∈ K
are redundant in the JNOB problem formulation in (2.12), but they are not necessarily au- tomatically satisfied in the associated continuous relaxation in (2.13) (cf. Section 2.3.1). As a result, adding the cuts in (2.12d) and (2.12e) into the continuous relaxation (2.13) can remove some non-integer solutions and tighten the continuous relaxation in (2.13). In addi- tion to such problem-specific cuts in (2.12e), there are also general cuts that are valid for all MISOCPs, like the Clique-cuts, and the Gomory-cuts [67–69, 81, 82].
Disconnected set Continuous relaxation Applying cuts
Figure 2.3: From left to right: the feasible set of a MISOCP, the feasible set of the associ- ated continuous relaxation, and the feasible set of the associated continuous relaxation after applying cuts.
The MIP solver IBM ILOG CPLEX implements the standard parallel BnC method [67– 69, 81, 82]. CPLEX offers users the full control of the BnC solution process, such as adding problem-specific cuts, and stopping the BnC tree-search when needed [81]. Control of the BnC solution process is the subject of various problem reformulations and customizing tech- niques discussed later in Section 2.4 and Section 2.5, respectively. The customizing strate- gies for the BnC method are part of the main contributions of this thesis and they will be
discussed in all the technical chapters. Moreover, the solver CPLEX also records the best- known GLB computed in the BnC procedure. The best-known GLB, or the best-known global upper bound (GUB) for maximization problems (see, e.g., Chapter 3), can be utilized to characterize the quality of the solutions found by CPLEX and to evaluate the performance of low-complexity heuristic algorithms.